On Minrank and Forbidden Subgraphs

TL;DR

This paper investigates the minrank of graphs with forbidden subgraphs in their complements, providing bounds for finite fields and explicit constructions over the reals, with implications for information theory and complexity.

Contribution

It establishes nearly tight bounds for minrank related to forbidden subgraphs over finite fields and constructs explicit graphs over reals, disproving a prior conjecture.

Findings

Lower bound of (\u221a n / \u2206 n) for triangle-free complements over finite fields

Explicit construction over reals showing minrank  n^ for non-bipartite graphs

Disproof of a conjecture by Codenotti, Pudle1k, and Resta

Abstract

The minrank over a field $\mathbb{F}$ of a graph $G$ on the vertex set $\{1,2,\ldots,n\}$ is the minimum possible rank of a matrix $M \in \mathbb{F}^{n \times n}$ such that $M_{i,i} \neq 0$ for every $i$, and $M_{i,j}=0$ for every distinct non-adjacent vertices $i$ and $j$ in $G$. For an integer $n$, a graph $H$, and a field $\mathbb{F}$, let $g(n,H,\mathbb{F})$ denote the maximum possible minrank over $\mathbb{F}$ of an $n$-vertex graph whose complement contains no copy of $H$. In this paper we study this quantity for various graphs $H$ and fields $\mathbb{F}$. For finite fields, we prove by a probabilistic argument a general lower bound on $g(n,H,\mathbb{F})$, which yields a nearly tight bound of $\Omega(\sqrt{n}/\log n)$ for the triangle $H=K_3$. For the real field, we prove by an explicit construction that for every non-bipartite graph $H$, $g(n,H,\mathbb{R}) \geq n^\delta$ for some $\delta = \delta(H)>0$. As a by-product of this construction, we disprove a conjecture of Codenotti, Pudl\'ak, and Resta. The results are motivated by questions in information theory, circuit complexity, and geometry.